Reciprocal Function f(x) = 1/x: Graph, Derivative, Integral and Limits

The reciprocal function f(x) = \frac{1}{x} sends each number to one over itself. It is the simplest rational function, the power function with exponent -1, and the standard example of infinite limits, asymptotes and the natural logarithm's derivative.

Formula

Graph of 1/x

The graph is a hyperbola with two branches, one in the first quadrant and one in the third.

Key properties

• Domain and range. All reals except 0, for both.
• Odd function. \frac{1}{-x} = -\frac{1}{x}, so the graph is symmetric about the origin.
• Asymptotes. Vertical asymptote x = 0, horizontal asymptote y = 0.
• Its own inverse. Solving y = \frac{1}{x} for x gives x = \frac{1}{y}, so f^{-1} = f and the graph is symmetric about y = x.
• Discontinuous only at 0, which is not in the domain. It is continuous on its domain.

Limits of 1/x

Because the one-sided limits at 0 disagree, \lim_{x \to 0}\frac{1}{x} does not exist. The limit at infinity is the basic building block for limits at infinity of every rational function.

Derivative of 1/x

This follows from the power rule with n = -1. The derivative is negative everywhere it is defined, so the function decreases on each branch, and the tangent lines get steeper near 0 and flatter far away. The second derivative is \frac{2}{x^3}: concave up for x > 0, concave down for x < 0.

Integral of 1/x

The reverse power rule would give \frac{x^0}{0}, which is undefined, so \frac{1}{x} is the one power that integrates to a natural logarithm. The absolute value covers negative x: for x < 0, \frac{d}{dx}\ln(-x) = \frac{-1}{-x} = \frac{1}{x}.

A definite integral cannot cross x = 0: \int_{-1}^{1}\frac{1}{x}\,dx is an improper integral that diverges.

Worked examples

Example 1: tangent line. At x = 2, f(2) = \frac{1}{2} and f'(2) = -\frac{1}{4}, so the tangent line is y = \frac{1}{2} - \frac{1}{4}(x - 2) = -\frac{1}{4}x + 1.

Example 2: definite integral. \int_1^e \frac{1}{x}\,dx = \ln e - \ln 1 = 1.

Example 3: a shifted reciprocal. \frac{d}{dx}\frac{3}{x - 1} = -\frac{3}{(x - 1)^2} and \int \frac{3}{x - 1}\,dx = 3\ln|x - 1| + C.

Common mistakes

1. Writing \int \frac{1}{x}\,dx = \frac{x^0}{0} or \ln x without the absolute value.
2. Getting the derivative sign wrong. \frac{d}{dx}\frac{1}{x} is -\frac{1}{x^2}, not \frac{1}{x^2}.
3. Confusing \frac{1}{x} with x^{-1} as an inverse function. \frac{1}{f(x)} is a reciprocal, while f^{-1}(x) usually means an inverse. For \frac{1}{x} they happen to agree, which is unusual.
4. Integrating across the asymptote as if the function were continuous there.

How it connects

The area under \frac{1}{t} from 1 to x is exactly \ln x, which is one way to define the natural logarithm. The power functions page covers the family x^n that \frac{1}{x} belongs to, and the harmonic series \sum \frac{1}{n} is its discrete cousin.